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\title[Robust Optimization]{Robust optimization (stability) of dynamical system}
\subtitle{closed-loop robust control approach}

\author[HU Jun]{Jun HU}
\institute{Orange Labs}
\date{\today}


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\begin{document}

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\begin{frame}[plain]
  \titlepage
\end{frame}

\begin{frame}
\frametitle{Content}
\tableofcontents
\end{frame}


\section{Case study}
\subsection{Dynamical system and uncertainty}
\begin{frame}
\frametitle{Case study}
\begin{itemize}
\item Types of uncertainty:
\begin{enumerate}
\item Periodic demands during daytime.
\item Unpredictable hotspots. 
\end{enumerate}
\vitem Consider SON as an interconnected dynamical system (self-organizing, self-healing etc.,)
%\vitem 
\end{itemize}
\end{frame}


\section{Methods}

\subsection{Approaches in literature}

\begin{frame}
\frametitle{Approaches in literature: static}
\begin{itemize}
\item Convex counterpart approach:
\begin{itemize}
\item Transform an important class of mini-max optimization problems into tractable convex optimization problems
\item Find a mini-max solution which minimizes the worst scenario.  
\item Conservative and static
\end{itemize} 

\vitem Stochastic optimization:
\begin{itemize}
\item Regard the uncertain parameter in the optimization problem as a random variable for which a given probability distribution is assumed. In the corresponding chance constrained formulation, the probability of a constraint violation is asked to be below a given confidence probability.
\item Find a solution which minimizes the current cost and expected cost for possible scenarios. 
\end{itemize} 

\end{itemize}
\end{frame}


\begin{frame}
\frametitle{Approaches in literature: dynamic}
\begin{itemize}
\item Recoverable robustness
\begin{itemize}
\item Similar to two-stage stochastic optimization, 
\item Find a pair $(x,A)$ where $x$ is solution, $A$ is a repair algorithm. When the uncertainty reveals, new solution can be found by applying $A(x)$. 
\item Converge slowly.
\end{itemize}
\vitem Online anticipatory stochastic optimization:
\begin{itemize}
\item Given a stochastic process to sample the future scenarios and an offline algorithm. 
\item At each time stamp, the offline algorithm carrying out a solution which minimizes the sampled future scenarios. 
\end{itemize}

\vitem Control theory -- closed-loop control and robust stability
%\begin{itemize}
%\item Open-loop optimal solution is not robust. 
%\item Closed-loop optimization attempts to obtain the robust stability of the system
%\end{itemize}
\end{itemize}
\end{frame}

%\subsection{Online anticipatory stochastic optimization}
%\begin{frame}
%\frametitle{Online anticipation stochastic optimization}
%\begin{itemize}
%\item The distribution of uncertainty is known or can be estimated through historical data. 
%\vitem 
%\end{itemize}
%\end{frame}

\section{Control vs Optimization}
\subsection{Closed-loop robust control}

\begin{frame}
\frametitle{Closed-loop robust control -- Feedback control}
\begin{itemize}
\item Dynamical system -- consider the network as a dynamical system, the system can be illustrated as: 
\vfill
\begin{center}
\includegraphics[width=.4\textwidth]{pic/inputoutput}
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\vitem Control vs. Optimization: consider the controller as decision variable of optimization problem, desired objectives are constraints on controlled closed-loop system. 

\end{itemize}
\end{frame}


\begin{frame}
\frametitle{State space expression}
\begin{itemize}
\item State space expression for discrete time invariant dynamical system:
\begin{align}
x(k+1)&=Ax(k)+Bu(k)+\omega(k) \\
u(k) &= Kx(k)\texttt{ under control law}
\end{align}
where $x$ is the state of the system, $u$ is control input and $\omega$ is disturbance. The matrix $A$ and $B$ are time independent and known, the objective is to find feedback matrix $K$ for controller. 
\vitem Formulate the system in the form of dissipative dynamical system
\begin{itemize}
\item \textbf{Dissipation inequality} is equivalent to Lyapunov inequality for open system.
\item \textbf{Tractability}, the problem can be solved in the form of LMI (Linear Matrix Inequality) by existing solvers.   
\end{itemize} 
\end{itemize}
\end{frame}


\subsection{Dissipative dynamical system}

\begin{frame}
\frametitle{Dissipative dynamical system}
\begin{itemize}
%\item Closed dynamical system (Lyapunov function), open dynamical system (Dissipation inequality)

\item Dissipation Inequality: the energy released from the system to the environment is less than the energy supplied from the environment to the system. Given a supply function $S$, there exists a storage function $V$ such that: 
\begin{equation}
V(x(K+1))-V(x(k))\leq S(k)
\end{equation} 
where
\begin{itemize}
\item Supply function: $S(k)$.
\vitem Storage function $V$: a system energy measurement function explicitly based on the system state $x$.
\end{itemize}

\vitem \textbf{Objective}: to find a state feedback controller which renders system dissipative with respect the storage function. 
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Network modeling}
\begin{columns}
\column{0.5\textwidth}
\begin{itemize}
\item Consider the network as an interconnected dynamical system which consists of $n$ subsystems, each with state of $x_i$.  
\vitem The users enter the network as random instants and locations, which can be considers as the energy from the environment to subsystems. 

\vitem The user can switch between the sites. 

\end{itemize}

\column{0.5\textwidth}
\begin{center}
\includegraphics[width=.7\textwidth]{pic/interconnected}
\end{center}
\end{columns}

\end{frame}



\subsection{Thermodynamic model}
\begin{frame}
\frametitle{Thermodynamic model}
\begin{itemize}
\item Consider the network in the form of thermodynamic model, the system receives a supply of user calling demands, and stores a part of this \textbf{disorder} and dissipates the remaining to the environment through the network optimization algorithm. 
\vitem The disorder can be measured by \textbf{Ectropy} (dual definition of Entropy) in the thermodynamic model proposed in \cite{Haddad2005thermodynamic} which represents the \underline{storage function} $V$. The \underline{supply function} $S$ is also given in the same paper.  
\vitem The dissipativity depends on the network control (optimization), the objective of this problem is to find a state feedback controller which renders system dissipative.  
\end{itemize}
\end{frame}


\subsection{$H_\infty$ synthesis} 
\begin{frame}
\frametitle{$H_\infty$ synthesis -- Robust stability}
\begin{itemize}
\item $H_\infty$ allows to design optimal controller under the restrictions:
\begin{itemize}
\item Performance is measured in closed loop system
\item Only one performance measure is allowed
\item Without involvement of structured time-varying/nonlinear uncertainties
\item Design LTI (Linear Time-Invariant) controller only
\end{itemize}
These restrictions may be respected in our case. 
\vitem The objective is to minimize the $\gamma$, where $\frac{\|x\|_2}{\|\omega\|_2}<\gamma$. The method is to prove that there exists the matrix $K$ which holds the relation between $K$ and $\gamma$ in the form of LMI (A convex programming problem can be solved by existing solvers). 
\end{itemize}
\end{frame}


\begin{frame}
\frametitle{Difficulties in problem resolution}
Given a state space expression in form of 
\begin{center}
\begin{equation}
x(k+1) = Ax(k) + Bu(k) +\omega(k)
\end{equation}
\end{center}

\begin{itemize}
\item Find the indicator to define the state $x_i$ of each subsystem $i$.
\vitem Linearize the system.
\vitem Define the state matrix $A$ and the optimization input matrix $B$. 
\end{itemize}
\end{frame}


\begin{frame}
\frametitle{References}
\bibliographystyle{abbrv}
\bibliography{robust}
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\end{document}